Compute the arc length function $s(t)={\displaystyle \underset{a}{\overset{t}{}}}||r^{\text{'}}(u)||du$ for $r(t)=($:$2\sqrt[\phantom{2}]{24}{t}^{\frac{1}{2}},lnt,12t$:$),$ and $a=1.$

$($Use symbolic notation and fractions where needed.$)$

$S=$ help $($fractions$)$